Having one or more remoteness distinctions in the past tense is reasonably common, particularly in Papua, parts of the Amazon and parts of Africa: http://wals.info/feature/66A (note that the map is quite sparse with data points). I'll give examples mostly from Papuan languages as I'm more familiar with those than the other areas. See e.g. The Papuan Languages of New Guinea by William A. Foley for examples and further sources on specific languages.
A common distinction is earlier today ("hodiernal") vs. before today. WALS says that if there is one well-defined cutoff point (rather than say, just two different past tenses that are used contrastively, and where the exact difference between the two depends on context) then it's almost always that, but I know of at least one example where this isn't the case, Marind (Marind(TNG), PNG), which distinguishes past through yesterday from events before yesterday.
The before today category is often divided further, e.g. into yesterday ("hesternal") vs. before yesterday. Enga (Engan(TNG), PNG) is an example of a language with such a system. Some languages have even more distinctions. Yimas (Ramu-Lower Sepik, PNG) distinguishes ealier today, yesterday, roughly 2-5 days ago and before roughly 5 days ago, and may additionally sometimes use the irrealis mood for legendary or highly remote past. Yagua (Peba-Yaguan, Peru) has five clearly grammaticalised tenses: a few hours ago, a day ago, roughly one week to one month ago, roughly one to two months up to one or two years ago and further distant or legendary.
Distance contrasts in future are also possible, though to my knowledge much less common, an example is Dani (West-TNG, Indonesia) which has two optional tense markers in the likely status, which distinguish a relatively near future from a far-flung future.
In addition to these absolute tenses, the relative tenses common on Papuan "medial verbs" may also show multiple distinctions. A simple simultaneous/sequential distinction is very common, but some languages, e.g. Naasioi (South Bougainville, PNG) further divide the sequential category into immediate sequential and delayed sequential, and a couple of languages, e.g. Telefol (Ok (TNG), PNG) further divide the delayed sequential into later and much later.